Abstract
Let H be a set of connected graphs, each of which has order at least three, and suppose that there exist infinitely many connected H-free graphs of minimum degree at least two and all except for finitely many of them have a 2-factor. In [J. Graph Theory, 64(2010), 250-266], we proved that if |H|<= 3, then one of the members in H is a star. In this article, we determine the remaining members of H and hence give a complete characterization of the pairs and triples of forbidden subgraphs.